OFFSET
1,1
COMMENTS
Perfect numbers (A000396) are a proper subset of this sequence. Weird numbers (A006037) are numbers whose proper divisors sum to more than the number, but no subset sums to the number.
Odd elements are rare: the first few are 8925, 32445, 351351, 442365; there are no more below 100 million. See A065235 for more details.
A065205(a(n)) = 1. - Reinhard Zumkeller, Jan 21 2013
LINKS
Giovanni Resta, Table of n, a(n) for n = 1..10000 (terms 1..200 from T. D. Noe, terms 201..5000 from Amiram Eldar)
EXAMPLE
Proper divisors of 20 are 1, 2, 4, 5 and 10. {1,4,5,10} is the only subset that sums to 20, so 20 is in the sequence.
MAPLE
filter:= proc(n)
local P, x, d;
P:= mul(x^d+1, d = numtheory:-divisors(n) minus {n});
coeff(P, x, n) = 1
end proc:
select(filter, [$1..2000]); # Robert Israel, Sep 25 2024
MATHEMATICA
okQ[n_]:= Module[{d=Most[Divisors[n]]}, SeriesCoefficient[Series[ Product[ 1+x^i, {i, d}], {x, 0, n}], n] == 1]; Select[ Range[ 1100], okQ] (* Harvey P. Dale, Dec 13 2010 *)
PROG
(Haskell)
a064771 n = a064771_list !! (n-1)
a064771_list = map (+ 1) $ elemIndices 1 a065205_list
-- Reinhard Zumkeller, Jan 21 2013
(Python)
from sympy import divisors
def isok(n):
dp = {0: 1}
for d in divisors(n)[:-1]:
u = {}
for k in dp.keys():
if (s := (d + k)) <= n:
u[s] = dp.get(s, 0) + dp[k]
if s == n and u[s] > 1:
return False
for k, v in u.items():
dp[k] = v
return dp.get(n, 0) == 1
print([n for n in range(1, 1039) if isok(n)]) # Darío Clavijo, Sep 17 2024
CROSSREFS
KEYWORD
nonn,nice
AUTHOR
Jonathan Ayres (jonathan.ayres(AT)btinternet.com), Oct 19 2001
EXTENSIONS
STATUS
approved